15: PCA in Finance

Principal Component Analysis is the SVD applied to data. In finance, it answers a fundamental question: what are the hidden factors driving asset returns? The math reveals that a few eigenvectors often explain most of the variance in hundreds of correlated assets.

The Setup

You have a returns matrix $X$ of shape $T \times n$:

• $T$ time periods (rows)
• $n$ assets (columns)
• Each entry $X_{ti}$ is the return of asset $i$ at time $t$

Assets are correlated. The goal: find uncorrelated factors that explain the variance.

PCA Algorithm

1. Center the data: Subtract the mean return from each column $\tilde{X}_{ti} = X_{ti} - \bar{X}_i$

2. Compute the covariance matrix: $\Sigma = \frac{1}{T-1} \tilde{X}^T \tilde{X}$

3. Eigendecompose: $\Sigma = V \Lambda V^T$

   • Columns of $V$ are principal components (eigenvectors)
   • Diagonal of $\Lambda$ are eigenvalues (variance explained)

4. Project data: $Z = \tilde{X} V$ gives factor exposures

Alternatively, use SVD directly: $\tilde{X} = U \Sigma V^T$. The right singular vectors $V$ are the principal components.

Interpretation in Finance

Principal Components as Factors

Each principal component $\mathbf{v}_i$ is a portfolio—a linear combination of assets. The first few PCs often have clear interpretations:

Equity markets (e.g., S&P 500 stocks):

• PC1: Market factor (all stocks move together)—explains ~50-70% of variance
• PC2: Often size or value (small vs. large, growth vs. value)
• PC3+: Sector or style factors

Fixed income (yield curve):

• PC1: Level (parallel shifts)—explains ~90% of variance
• PC2: Slope (steepening/flattening)
• PC3: Curvature (butterfly)

Variance Explained

The eigenvalue $\lambda_i$ is the variance of the $i$-th principal component. The fraction of variance explained:

The cumulative variance explained tells you how many PCs you need:

If the first 3 PCs explain 80% of variance, the other $n-3$ dimensions are mostly noise.

Dimensionality Reduction

Factor Models

Instead of modeling $n$ correlated returns, model $k \ll n$ uncorrelated factors:

where:

• $\mathbf{f}_t$ is the $k$-dimensional factor return vector (the PCs)
• $B$ is the $n \times k$ factor loading matrix (columns of $V_k$)
• $\boldsymbol{\epsilon}_t$ is idiosyncratic noise

This is the statistical factor model. It reduces a $n \times n$ covariance matrix to:

where $D$ is diagonal (idiosyncratic variances).

Benefits

1. Fewer parameters: Estimating $n(n+1)/2$ covariances vs. $nk + k$ factor loadings
2. More stable: Less overfitting to noise
3. Interpretable: Factors often have economic meaning

Example: Yield Curve PCA

Consider monthly changes in Treasury yields at maturities: 3M, 6M, 1Y, 2Y, 3Y, 5Y, 7Y, 10Y, 20Y, 30Y.

Typical results:

PC1 loadings: All positive, roughly equal—a parallel shift.

PC2 loadings: Negative for short maturities, positive for long—a steepening/flattening.

PC3 loadings: Positive at short and long ends, negative in the middle—a butterfly.

Three numbers describe 99%+ of yield curve movements!

Risk Management Applications

Factor Risk Decomposition

Portfolio variance decomposes by factor:

The term $(\mathbf{w}^T \mathbf{v}_i)^2 \lambda_i$ is the risk contribution from factor $i$.

Stress Testing

To stress test against a factor shock:

1. Identify the relevant PC (e.g., PC1 for market crash)
2. Compute portfolio exposure: $\beta_i = \mathbf{w}^T \mathbf{v}_i$
3. Multiply by shock size: $\Delta P = \beta_i \times \text{shock}$

De-correlating Portfolios

Transform returns to PC space: $\mathbf{z}_t = V^T \mathbf{r}_t$. Now:

• Components are uncorrelated
• Variances are eigenvalues
• Risk budgeting becomes simple

Practical Considerations

Stationarity

PCA assumes the covariance structure is stable. In reality:

• Correlations spike during crises
• Factor structure can change over time

Solution: Rolling-window PCA, regime-switching models, or robust covariance estimators.

Sign and Scale Ambiguity

Eigenvectors are determined up to sign. $\mathbf{v}$ and $-\mathbf{v}$ are both valid. Choose signs for interpretability (e.g., PC1 loadings all positive for “market”).

Number of Factors

How many PCs to keep? Common approaches:

• Scree plot: Look for an “elbow” in eigenvalue decay
• Variance threshold: Keep enough for 80-90% cumulative variance
• Cross-validation: Test predictive performance

PCA vs. Factor Models

PCA finds factors that maximize variance explained. Economic factors are chosen for interpretability. Often they overlap significantly (PC1 ≈ market factor).

Key Takeaways

1. PCA = eigendecomposition of covariance: Finds uncorrelated directions of maximum variance
2. Few factors explain most variance: Especially in correlated markets
3. Interpretable in finance: Level/slope/curvature for rates, market/size/value for equities
4. Enables dimensionality reduction: Model $k$ factors instead of $n$ assets
5. Risk decomposes by factor: Understand where portfolio risk comes from

When you see 500 stocks moving in sync, you’re seeing PC1—the market factor. PCA reveals that behind the apparent complexity of financial markets, a handful of common factors drive most of the action. The rest is noise.